Internal semantics of a category based on a fixed topos.

There is no good way of doing this. For instance, consider a theory $\mathbb{T}$ such that $\mathbb{T}$-models in $\mathbf{Set}$ do not exist. Clearly, going by the category of models in $\mathbf{Set}$ alone, it will be impossible to distinguish $\mathbb{T}$ from the inconsistent theory. There are more sophisticated versions of this phenomenon. For example, the category of complete join semilattices is indistinguishable from the category of $\kappa$-complete join semilattices, where $\kappa$ is the size of the universe. In order to avoid this issue, we must restrict to a fragment of logic for which we have a sufficiently strong completeness theorem, such as coherent logic.

So let $\mathbb{T}$ be a coherent theory and let $\mathcal{M}$ be the category of models of $\mathbb{T}$ in $\mathbf{Set}$. Then $\mathcal{M}$ is an accessible category with colimits of filtered diagrams (but not every such category is the category of models for a coherent theory). Since $\mathbb{T}$ is coherent, there is a category $\mathcal{B}$ together with a set $\mathcal{L}$ of cones over finite diagrams in $\mathcal{B}$ and a set $\mathcal{C}$ of cocones under finite diagrams in $\mathcal{B}$ such that $\mathcal{M}$ is equivalent to the category of functors $\mathcal{B} \to \mathbf{Set}$ that send members of $\mathcal{L}$ to limiting cones and members of $\mathcal{C}$ to colimiting cocones. (Moreover, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{L}$ are canonical given $\mathbb{T}$.) The category of $\mathbb{T}$-models in a topos $\mathcal{E}$ is defined analogously: it is the category of functors $\mathcal{B} \to \mathcal{E}$ such that send members of $\mathcal{L}$ to limiting cones and members of $\mathcal{C}$ to colimiting cocones.

In the case where $\mathcal{M}$ is the category of models for a Horn theory (or, more generally, a cartesian theory), there is a canonical choice of $\mathbb{T}$ and hence a canonical choice of $\mathcal{B}$, $\mathcal{L}$, and $\mathcal{C}$: namely, $\mathcal{B}^\mathrm{op}$ is a skeleton of the category of finitely presented models, $\mathcal{L}$ is the set of all limiting cones over finite diagrams in $\mathcal{B}$, and $\mathcal{C}$ is empty. This covers the case where $\mathcal{M}$ is the category of models of an algebraic theory.